A Nonlocal p-laplacian Equation With Variable Exponent For Image Restoration

Transcription

1 A Nonlocal p-laplacian Equation With Variable Exponent For Image Restoration EST Essaouira-Cadi Ayyad University SADIK Khadija Work with : Lamia ZIAD Supervised by : Fahd KARAMI Driss MESKINE Premier congrès Franco Marocain de Mathématiques appliquées, April, Marrakech Morocco 20 April 2018

2 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) 5 2 / 38

3 Image Processing Image processing : processing of images using mathematical operations convert an image into digital form and apply some operations on it. get an enhanced image extract some useful information from image. EDGES TEXTURES 3 / 38

4 Minimisation The restored image u is computed from the following minimization : mine(u) = Ψ(x, u )dx + λ u 2 f u 2 L 2 (Ω), (1) Ω }{{} Regularizing term }{{} Fidelity term 4 / 38

5 Minimisation The restored image u is computed from the following minimization : mine(u) = Ψ(x, u )dx + λ u 2 f u 2 L 2 (Ω), (1) Ω }{{} Regularizing term }{{} Fidelity term Case Ψ(x, s) = s2 2 The smoothing Method (Gabor) [1] (1960). 4 / 38

6 Minimisation The restored image u is computed from the following minimization : mine(u) = Ψ(x, u )dx + λ u 2 f u 2 L 2 (Ω), (1) Ω }{{} Regularizing term }{{} Fidelity term Case Ψ(x, s) = s2 2 The smoothing Method (Gabor) [1] (1960). Case Ψ(x, s) = log(1+s2 ) 2 Perona-Malik [2] (1990). 4 / 38

7 Minimisation The restored image u is computed from the following minimization : mine(u) = Ψ(x, u )dx + λ u 2 f u 2 L 2 (Ω), (1) Ω }{{} Regularizing term }{{} Fidelity term Case Ψ(x, s) = s2 2 The smoothing Method (Gabor) [1] (1960). Case Ψ(x, s) = log(1+s2 ) 2 Perona-Malik [2] (1990). Case Ψ(x, s) = s Total Variation (ROF) [3] (1992). 4 / 38

8 Minimisation The restored image u is computed from the following minimization : mine(u) = Ψ(x, u )dx + λ u 2 f u 2 L 2 (Ω), (1) Ω }{{} Regularizing term }{{} Fidelity term Case Ψ(x, s) = s2 2 The smoothing Method (Gabor) [1] (1960). Case Ψ(x, s) = log(1+s2 ) 2 Perona-Malik [2] (1990). Case Ψ(x, s) = s Total Variation (ROF) [3] (1992). Case Ψ(x, s) = sp p p-laplacian equation [4] (2012). 4 / 38

9 The p-laplacian equation has been proved to be very powerful in the field of image denoising [4, 5]. u t div ( u p 2 u ) + λ(u f ) = 0 in Q := (0, T ) Ω, u. n = 0 in Σ := (0, T ) Γ, u 0 = f in Ω. (2) where p 1, u is the restored image and λ is a regularization parameter. 5 / 38

10 DrawBack!!!!! This model is very performing in removing noise. However, smaller details, such as texture, are destroyed. Noisy image local p-laplacian zoom of the result 6 / 38

11 Non local Non Local Means is introduced by Buades al in [9] (2005). The distance between patches : d a(b(p), B(q)) = The weight function is given by : ( da(b(p), B(q)) ) J(p, q) = exp, h 2 Ω G a(z) f (p + z) f (q + z) 2 dz, 7 / 38

12 Non local Non Local Means is introduced by Buades al in [9] (2005). The distance between patches : d a(b(p), B(q)) = The weight function is given by : ( da(b(p), B(q)) ) J(p, q) = exp, h 2 { J(p, q) 1 if B(p) and B(q) are similar, J(p, q) 0 if B(p) and B(q) aren t similar. Ω G a(z) f (p + z) f (q + z) 2 dz, 7 / 38

13 Kindermann, Osher and Jones (2005) [6] have proposed the nonlocal p-laplacian problem for deblurring and denoising images. Non local PDE and p Laplacian Equation Graphs Elmoataz (2012). The evolutionary nonlocal p-laplacian problem with homogeneous Neumann boundary conditions is described as : { u t (x) = J(x, y) u(y) u(x) p 2 (u(y) u(x))dy λ(f (x) u(x)) in Q, Ω u 0 = f in Ω. (3) Where J is the weight function. DrawBack : execution time!!!!! 8 / 38

14 The p(x)-laplacian equation was boosted by its application into various phenomena such as electrorheological and thermorheological fluids( Růžička [10]). And more recently, in the context of image processing, Chen.al [11] and Aboulaich. al [12] have been suggested. u t div ( u p(x) 2 u ) + λ(u f ) = 0 in Q, u. n = 0 in Σ u 0 = f in Ω. (4) where p(x) = 1 + and λ is a regularization parameter. 1, k > 0, σ > 0, 1 + k G σ f 2 9 / 38

15 The main gaol 1 Using the non local p-laplacian operators in order to seek the information in the most resembling structures (patches). 2 Using the variable exponent. The non local p-laplacian with variable exponent. NLPx-Laplacian 10 / 38

16 The nonlocal variable exponent p(x)-evolution reaction diffusion equation can be written as : u (P) t(x) = J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x))dy λ(f (x) u(x)) in Q, Ω u(0, x) = u 0 (x) = f (x), in Ω. (5) 11 / 38

17 The nonlocal variable exponent p(x)-evolution reaction diffusion equation can be written as : u (P) t(x) = J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x))dy λ(f (x) u(x)) in Q, Ω u(0, x) = u 0 (x) = f (x), in Ω. (5) where p : Ω Ω (1, ) is a continuous symmetric variable exponent in Ξ = Ω Ω such that p(x, y) = p(y, x), p (x) = inf y Ω p(x, y) and p + (x) = sup y Ω p(x, y). (6) 11 / 38

18 The nonlocal variable exponent p(x)-evolution reaction diffusion equation can be written as : u (P) t(x) = J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x))dy λ(f (x) u(x)) in Q, Ω u(0, x) = u 0 (x) = f (x), in Ω. (5) where p : Ω Ω (1, ) is a continuous symmetric variable exponent in Ξ = Ω Ω such that p(x, y) = p(y, x), p (x) = inf y Ω p(x, y) and p + (x) = sup y Ω p(x, y). (6) And the kernel J : R N R N R is a nonnegative symmetric continuous smooth function with compact support contained in Ω B(0, d) R N R N such that 0 < sup J(x, y) = R(x) L (Ω), and y B(0,d) J(x, y)dx = 1 R N (7) 11 / 38

19 Existence and uniqueness of a solution. For δ > 0, we consider ( t k a δ discretization of [0, T ), that is )k=1,...n t 0 = 0 < t 1 <... < t n 1 < T = t n. For any δ > 0, we say that u δ is a δ approximate solution of (P), if there exists ( t k a δ discretization )k=1,...n of [0, T ), such that u 0 for t [0, t 1 ], u δ (t) = (8) for t ]t k 1, t k ], k = 2,...n u k 12 / 38

20 Existence and uniqueness of a solution. and for k = 1,..., n, the functions u k solves the Euler implicit time discretization of (P) : u k (x) δ J(x, y) u k (y) u k (x) p(x,y) 2 (u k (y) u k (x))dy +δλ(u k (x) f (x)) = u k 1 (x). Ω See that the generic problem is given by : v(x) r Ω J(x, y) v(y) v(x) p(x,y) 2 (v(y) v(x))dy + rλv(x) = φ, where r 1 is a given constant and φ :: Ω R is a given application. (9) 13 / 38

21 Existence and uniqueness Theorem Let f L 2 (Ω),( λ > 0 and p, ) q satisfies ((6)-(7). Then) there exists a unique solution u C [0, T ]; L 1 (Ω) W 1,1 [0, T ]; L 1 (Ω) of problem (P) satisfies u(0, x) = u 0 (x), a.e. x Ω and u t(t, x) = J(x, y) u(t, y) u(t, x) p(x,y) 2 (u(t, y) u(t, x))dy 2λ(u(t, x) f (x)) in Q, Ω Moreover, if u δ is a δ- approximate solution, then ( ) u δ u in C [0, T ]; L 1 (Ω) as δ / 38

22 Existence and uniqueness Proving the existence of the solution to the problem (P) using Semigroup Theory. Step 1 : We define the operator L λ : L λ u(x) = J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x))dy+λu(x), x Ω Ω 15 / 38

23 Existence and uniqueness Proving the existence of the solution to the problem (P) using Semigroup Theory. Step 1 : We define the operator L λ : L λ u(x) = J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x))dy+λu(x), x Ω Ω Step 2 : L λ is completely accretive. 15 / 38

24 Existence and uniqueness Proving the existence of the solution to the problem (P) using Semigroup Theory. Step 1 : We define the operator L λ : L λ u(x) = J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x))dy+λu(x), x Ω Ω Step 2 : L λ is completely accretive. Step 3 : Prove that L λ is verifies a range condition. 15 / 38

25 Step 1 : Operator definition Definition Defining in L 1 (Ω) the operator L λ by L λ u(x) = J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x))dy+λu(x), x Ω. Ω 16 / 38

26 Moreover, we have the following integration by parts lemma. Lemma For every u, ξ L p (Ω), we have J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x))dyξ(x)dx Ω Ω = 1 J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x))(ξ(y) ξ(x))dydx. 2 Ω Ω (10) 17 / 38

27 Step 2 Lemma The operator L λ is completely accretive. ( ) L λ u(x) L λ v(x) S(u(x) v(x))dx Ω = 1 ( J(x, y) u(y) u(x) p(x,y) 2 (u(y) u(x)) 2 Ω Ω ) v(y) v(x) p(x,y) 2 (v(y) v(x)) dy ( S(u(y) v(y)) S((u(x) v(x)) ) dx + λ (u(x) v(x))s(u(x) v(x))dx 0. (11) Ω 18 / 38

28 Step 3 : L λ verifies a range condition Proposition Assume that the conditions (6)-(7) are satisfied, for any φ L 2 (Ω), there exists a unique solution u L p (Ω) of the problem (9) satisfying v(x) J(x, y) v(y) v(x) p(x,y) 2 (v(y) v(x))dy+λv(x) = φ a.e. in Ω. Ω 19 / 38

29 Proof of Theorem 1 It is easy to see that, D(L λ ) = L 1 (Ω). Thanks to the proposition (1) and the the general Nonlinear Semi-group Theory the problem (PS) { ut (t) + L λ (u(t)) = λf in (0, T ), u(0) = u 0, has a unique mild solution u for every initial data u 0 and f in L 1 (Ω), and ( ) u δ u in C [0, T ]; L 1 (Ω) as δ 0, where u δ is a δ- approximate solution given by (8)-(9). Using the complete accretivity of the operator L λ the solution u(t) of the problem (PS) is a solution of (P) in the sense of Theorem / 38

30 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) The numerical results The discrete iterative schemes of the problem (P) using the explicit Euler method can be written as : u k+1 i ui k = J ij uj k ui k pij 2 (uj k ui k ) λ(f i ui k ), τ (12) j N i ui 0 = f i, where N i is the neighbors set, τ is the time step size and k is the iteration number. 21 / 38

31 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) The weights function is given by : ( J(i, j) = exp d a(b(i), B(j)) ) σ 2, where d a (B(i), B(j)) = G a (z) f (i + z) f (j + z) 2 dz, Ω For the numerical expermiments, we choose a patches size of11 11 (i.e. P = 5), a search window of (i.e.n w = 11) and σ = / 38

32 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) The variable exponent is given by : where p i = 1 + p ij = p i + p j, k G µ f i 2. For the numerical expermiments, we choose µ = 0.5 andk = / 38

33 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) PSNR = PSNR = λ = 0.01,dt = 0.1, and d = / 38

34 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) PSNR = PSNR = λ = 0.01,dt = 0.1, and d = / 38

35 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) PSNR = PSNR = λ = 0.01,dt = 0.1, and d = / 38

36 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) 27 / 38

37 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) 28 / 38

38 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) 29 / 38

39 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) Noisy image local p-laplacian Nonlocal p-laplacian 30 / 38

40 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) Zoom of Local Zoom of Nonlocal 31 / 38

41 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) Stopping criteria : PSNR = Noisy image PSNR= Results after 26 iterations Result of NLPL PSNR= Result of NLPxL PSNR= / 38

42 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) Stopping criteria : PSNR = Noisy image Why PSNR= 30.87??? Result of NLPL 310 iterations with s Result of NLPxL 26 iterations s 33 / 38

43 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) Noisy image PSNR= Result of NLPL iterations. Result of NLPxL PSNR= iterations. 34 / 38

44 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) What about Textures??? Zoom of NLPL Zoom of NLPxL 35 / 38

45 Efficiency Test Comparison Test(local Vs NonLocal) Comparison Test(NLPL Vs NLPxL) 36 / 38

46 M. Lindenbaum, M. Fischer, and A. M. Bruckstein, On Gabor s contribution to image enhancement. Pattern Recognition, 27(1) :1 8, P. Perona, J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell, (1990), 12(7), L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), pp W. Wei, B. Zhou, A p-laplace Equation Model for Image Denoising, Information Technology Journal, (2012), vol.11, Issue 5, E. Dibendetto, Degenerate Parabolic Equations, Spring-Verlag, New York S. Kindermann, S. Osher, P.W. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Simul., 4 (2005), 37 / 38

47 Thank you for your attention 38 / 38